Mathematics Dictionary Dr. K. G. Shih

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Integral Table

Questions

Read Symbol defintion Q01 | - Derivative : Polynomial and rational functions Q02 | - Derivative : Trigonometric functions Q03 | - Derivative : Exponential family Q04 | - Derivative : Logarthemic familay Q05 | - Integral : Polynomial and rational functions Q06 | - Integral : Trigonometrical functions Q07 | - Integral : Exponential family Q08 | - Integral : Logarithmic family Q09 | - Q10 | -

Answers

Q01. Derivative : Polynomial and rational functions Power with positve or negative integer d/dx(x^n) = n*x^(n-1) d/dx(x^(-1)) = -1*x^(-2) d/dx(x^(-2)) = -2*x^(-3) Rational d/dx(1/(1 + x^2) = arctan(x) Irrational d/dx(x^(+1/2)) = (+1/2)*(x^(-1/2) d/dx(x^(-1/2)) = (-1/2)*(x^(-3/2) d/dx(Sqr(x)) = d/dx(x^(+1/2)) = (+1/2)*(x^(-1/2) d/dx(1/Sqr(1-x^2) = arcsin(x) Chain rule Example : If y = u^5, find dy/dx Wrong answer dy/dx = d/dx(u^5) = 5*u^4 y is function of u d/dx is with respect x not to u We can not use power rule in this case. Correct answer : Use chain rule d/dx(u^5) = (d/du(u^5))*du/dx d/dx(u*5) = (4*u^4)*(du/dx) Go to Begin Q02. Derivative : Trigonometric functions d/dx(sin(x)) = +cos(x) d/dx(cos(x)) = -sin(x) d/dx(tan(x)) = +sec(x)^2 d/dx(arcsin(x)) = +1/Sqr(1 - x^2) d/dx(arccos(x)) = -1/Sqr(1 - x^2) d/dx(arctan(x)) = +1/(1 + x^2) Go to Begin Q03. Derivative : Exponential familay d/dx(e^x) = e^x d/dx(sinh(x)) = +cosh(x) d/dx(cosh(x)) = +sinh(x) d/dx(tanh(x)) = +sech(x)^2 d/dx(csch(x)) = -csch(x)*coth(x) d/dx(sech(x)) = -sech(x)*tanh(x) d/dx(coth(x)) = +csch(x)^2 Go to Begin Q04. Deriative : Logarithic family d/dx(ln(x)) = 1/x d/dx(arcsinh(x)) = +1/Sqr(x^2 + 1) d/dx(arccosh(x)) = +1/Sqr(x^2 - 1) d/dx(arctanh(x)) = +1/(1 - x^2) d/dx(arccsch(x)) = +1/(x*Sqr(x^2 + 1)) d/dx(arcsech(x)) = +1/(x*Sqr(x^2 - 1)) d/dx(arccoth(x)) = -1/(1 - x^2) Go to Begin Q05. Integral : Polynomial and rational functions ∫ (x^n)dx = (x^(n+1))/(n+1) + C and n NE -1 ∫ (1/(1+x^2))dx = arctan(x) Go to Begin Q06. Intgral : Trigonometrical function Function ∫sin(x)dx = -cos(x) + C ∫cos(x)dx = +sin(x) + C ∫tan(x)dx = -ln(cos(x) + C ∫csc(x)dx = +ln(csc(x) - cot(x)) + C ∫sec(x)dx = +ln(sec(x) + tan(x)) + C ∫cot(x)dx = +ln(sin(x)) + C Inverse function ∫arcsin(x)dx = +1/Sqr(1 - x^2) + C ∫arccos(x)dx = -1/Sqr(1 - x^2) + C ∫arctan(x)dx = +1/(1 + x^2) + C ∫arccsc(x)dx = +1/(x*Sqr(x^2 - 1)) + C ∫arcsec(x)dx = -1/(x*Sqr(x^2 - 1)) + C ∫arccot(x)dx = -1/(1+x^2) + C Go to Begin Q07. Integral : Exponential family ∫exp(x)dx = exp(x) + C ∫sinh(x)dx = cosh(x) + C ∫cosh(x)dx = sinh(x) + C ∫tanh(x)dx = ln(cosh(x)) + C ∫csch(x)dx = ln(tanh(x/2)) + C ∫sech(x)dx = arctan(sinh(x)) + C ∫coth(x)dx = ln(sinh(x) + C Go to Begin Q08. Integral : Logaithmic familay ∫arcsinh(x)dx = x*arcsinh(x) - Sqr(x^2 + 1) ∫arccosh(x)dx = x*arccosh(x) - Sqr(x^2 - 1) ∫arctanh(x)dx = x*arctanh(x) + ln(1 - x^2)/2 ∫arccsch(x)dx = x*arccsch(x) + arcsinh(x) ∫arcsech(x)dx = x*arcsech(x) - arccos(z) ∫arccoth(x)dx = x*arccoth(x) + ln)/(x^2 - 1)/2 Go to Begin Q09. Answer Go to Begin Q10. Answer Go to Begin

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